A Residual Inverse Power Method

Abstract

The inverse power method involves solving shifted equations of the form $(A -\sigma I)v = u$. This paper describes a variant method in which shifted equations may be solved to a fixed reduced accuracy without affecting convergence. The idea is to alter the right-hand side to produce a correction step to be added to the current approximations. The digits of this step divide into two parts: leading digits that correct the solution and trailing garbage. Hence the step can be be evaluated to a reduced accuracy corresponding to the correcting digits. The cost is an additional multiplication by $A$ at each step to generate the right-hand side. Analysis and experiments show that the method is suitable for normal and mildly nonnormal problems.